The value of speed of light in different regions of spacetime

Question

This question of mine started shaping in my head first while I was looking for the most fundamental answer for the speed of light's value and its property of being the limit.

I have convinced myself that it's due to the structure of spacetime, or in other words because the spacetime interval is expressed as $ds^2=dx^2+dy^2+dz^2-c^2dt^2$ which is not Euclidean (for the limit property part) and has the value $9\times10^{10}$ in front of the $dt^2$ term (for the value part).

So from this conclusion, and by knowing the fact that the coefficients of those coordinate terms can change with a "curvature" in spacetime, I wonder if the speed of light can have some other value in different curved regions in spacetime.

I don't mean it as differing between different observers in that same curved region but as differing between different observers in different regions with different curvatures.

Answer 1

It is not a change in curvature that precipitates a change in the coefficients of the metric, but a change of coordinates. Indeed, at any event in any spacetime (regardless of how strongly curved it is), there are coordinate systems where the coefficients are all $\pm 1$.

What curvature does is prevent extending such coordinate systems. These inertial coordinates are only valid locally.

Similarly, even in flat spacetime the metric coefficients can be changed arbitrarily at a given event by simply changing the coordinates appropriately. This can change the speed of light to something other than $c$. Note: this does not violate the second postulate because the second postulate applies only to inertial frames.